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A319826 GCD of the strict integer partition with FDH number n; GCD of the indices (in A050376) of Fermi-Dirac prime factors of n.

%I A319826 #10 Feb 18 2023 08:08:54
%S A319826 0,1,2,3,4,1,5,1,6,1,7,1,8,1,2,9,10,1,11,1,1,1,12,1,13,1,2,1,14,1,15,
%T A319826 1,1,1,1,3,16,1,2,1,17,1,18,1,2,1,19,1,20,1,2,1,21,1,1,1,1,1,22,1,23,
%U A319826 1,1,3,4,1,24,1,2,1,25,1,26,1,1,1,1,1,27,1,28
%N A319826 GCD of the strict integer partition with FDH number n; GCD of the indices (in A050376) of Fermi-Dirac prime factors of n.
%C A319826 Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1, ..., y_k) is f(y_1) * ... * f(y_k).
%H A319826 Antti Karttunen, <a href="/A319826/b319826.txt">Table of n, a(n) for n = 1..16384</a>
%F A319826 For all n >= 1, a(A050376(n)) = n. - _Antti Karttunen_, Feb 18 2023
%e A319826 45 is the FDH number of (6,4), which has GCD 2, so a(45) = 2.
%t A319826 nn=200;
%t A319826 FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
%t A319826 FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
%t A319826 GCD@@@Table[Reverse[FDfactor[n]/.FDrules],{n,nn}]
%o A319826 (PARI) A319826(n) = { my(i=1,g=0,x=A052331(n)); while(x,if(x%2,g = gcd(g,i)); x>>=1; i++); (g); }; \\ (Uses the program given in A052331) - _Antti Karttunen_, Feb 18 2023
%Y A319826 A left inverse of A050376.
%Y A319826 Cf. A052331, A056239, A064547, A213925, A289508, A289509, A290103, A299755, A299757, A319825.
%K A319826 nonn
%O A319826 1,3
%A A319826 _Gus Wiseman_, Sep 28 2018
%E A319826 Secondary definition added by _Antti Karttunen_, Feb 18 2023